Problem: A circle has a circumference of $18\pi$. It has an arc of length $\dfrac{21}{4}\pi$. What is the central angle of the arc, in degrees? ${18\pi}$ ${105^\circ}$ $\color{#DF0030}{\dfrac{21}{4}\pi}$
Answer: The ratio between the arc's central angle $\theta$ and $360^\circ$ is equal to the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{360 ^ \circ} = \dfrac{s}{c}$ $\dfrac{\theta}{360 ^ \circ} = \dfrac{21}{4}\pi \div 18\pi$ $\dfrac{\theta}{360 ^ \circ} = \dfrac{7}{24}$ $\theta = \dfrac{7}{24} \times 360 ^ \circ$ $\theta = 105^\circ$